We present results of numerical simulations of sequences of binary-single scattering events of black holes in dense stellar environments. The simulations cover a wide range of mass ratios from equal mass objects to 1000:10:10 solar masses and compare purely Newtonian simulations with a relativistic endpoint, simulations in which Newtonian encounters are interspersed with gravitational wave emission from the binary, and simulations that include the effects of gravitational radiation reaction by using equations of motion that include the 2.5-order post-Newtonian force terms, which are the leading-order terms of energy loss from gravitational waves. In all cases, the sequence is terminated when the binary's merger time due to gravitational radiation is less than the arrival time of the next interloper. We also examine the role of gravitational waves during an encounter and show that close approach cross-sections for three 1-solar-mass objects are unchanged from the purely Newtonian dynamics except for close approaches smaller than 0.00001 times the initial semimajor axis of the binary. We also present cross-sections for mergers resulting from gravitational radiation during three-body encounters for a range of binary semimajor axes and mass ratios including those of interest for intermediate-mass black holes (IMBHs). We find that black hole binaries typically merge with a very high eccentricity --- extremely high when gravitational waves are included during the encounter such that when the gravitational waves are detectable by LISA, most of the binaries will have eccentricities e > 0.9 though all will have circularized by the time they are detectable by LIGO. We also investigate the implications for the formation and growth of IMBHs and find that the inclusion of gravitational waves during the encounter results in roughly half as many black holes ejected from the host cluster for each black hole accreted onto the growing IMBH. The simulations show that the Miller & Hamilton model of IMBH formation is a viable method if it is modified to start with a larger seed mass
